Question

a weather balloon is 529 meters above the ground. two weather stations, a and b, monitor the balloons progress. compare the measures of the different angles of elevation and depression that are labeled in the diagram. which statements are true? check all that apply. x>y v>x w = y y<v v = w x<w

Explanation:

Step1: Recall the relationship between angle of elevation and depression

Angle of elevation from a point on the ground to the balloon is equal to the angle of depression from the balloon to that point. So \(v = x\) (angle of depression \(v\) and angle of elevation \(x\) for station \(A\)), \(w = y\) (angle of depression \(w\) and angle of elevation \(y\) for station \(B\)).

Step2: Use the tangent function

For angle \(x\), \(\tan x=\frac{529}{229}\approx2.31\). For angle \(y\), \(\tan y = \frac{529}{347}\approx1.52\). Since the tangent function \(y = \tan\theta\) is an increasing function for \(\theta\in(0,\frac{\pi}{2})\), and \(229<347\), we have \(x > y\).

Step3: Analyze the sum of angles

We know that \(v + w=180^{\circ}\) (linear - pair of angles). Since \(v=x\) and \(w = y\), and \(x>y\), we also know that \(y y\)) and \(xy\) and \(w = y\)), \(v>x\) is wrong (\(v = x\)), \(v = w\) is wrong (\(v=x\), \(w = y\) and \(x>y\))

Answer:

\(x>y\), \(w = y\), \(y